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lcm(y,z)=n

Source: Canadian Repêchage 2013: Problem 3

May 20, 2014
number theoryleast common multiplenumber theory proposed

Problem Statement

A positive integer nn has the property that there are three positive integers x,y,zx, y, z such that lcm(x,y)=180\text{lcm}(x, y) = 180, lcm(x,z)=900\text{lcm}(x, z) = 900, and lcm(y,z)=n\text{lcm}(y, z) = n, where lcm\text{lcm} denotes the lowest common multiple. Determine the number of positive integers nn with this property.
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